The $li_0$-$li_1$ bounds conjecture for the prime-counting function

Let $\pi(x)$ denote the number of primes at most $x$. Let $li_0(x)$ and $li_1(x)$ be the two truncated asymptotic approximation functions defined in the paper.

The $li_0$-$li_1$ bounds conjecture. For every real $x\ge e$,

$$li_0(x)\le \pi(x)\le li_1(x).$$

The paper motivates this conjecture using numerical comparisons and proved bounds over finite ranges. It also states that either this conjecture or the corresponding conjecture using $\underline{li}$ and $\overline{li}$ would imply the Riemann Hypothesis; the source does not establish the conjecture.